Binary numeral system
The binary number system, also referred to as base2, or radix2, represents numbers using only the digits 0 and 1. This is in contrast with the more familiar decimal numeral system (a.k.a. base10, radix10) which uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In the binary system, each digit position represents a power of two. The numeral "" in binary represents the value consisting of one set of twos () and no sets of ones (), which we are accustomed to seeing represented as "2". This is analogous to the decimal system, where each digit position represents a power of ten: the numeral "", for example, represents the value consisting of one set of tens (), and no sets of ones (). When the numeral system used for a number is in question, one can write the radix as a subscript to the number, as is done in the following table:
Binary  

Decimal 
Binary arithmetic
Arithmetic with binary numerals is similar to arithmetic with decimal numerals, except that the addition and multiplication tables are much simpler:


Division and subtraction are performed in the same way as for decimal numerals, but using the corresponding rules for binary addition and multiplication. Noninteger quantities can be represented as binary digits to the right of the binary point. For example,
Repeating binary expansions also occur, for any fraction where the denominator is not a power of 2. For example, (with 0011 repeating).
Irrational numbers can also be expressed, and will have irregular distributions of digits. For example,
Use in computing
The binary system is used in most electronic computers, as the values of 0 and 1 can be easily represented by a low and a high voltage in a circuit (i.e., by an "on/off" switch). A single digit of a binary numeral is referred to as a bit, short for binary digit. (The term bit was coined in 1947 at Bell Laboratories.) A bit can be a measure of data size, or a measure of information entropy, which are often not equal in size.
Other representations
Because the number of digits in the binary representation of a value can grow quickly, when human readability is desired binary values are often represented in the octal numeral system (base 8) or the hexadecimal numeral system (base 16). Octal uses the digits 0 through 7, while hexadecimal uses the digits 0 through 9, followed by the letters A through F to represent the values ten, eleven, twelve, thirteen, fourteen, and fifteen.
Binary numerals can be converted to octal by grouping the binary digits in groups of three beginning at the ones place, with each group of three binary digits converting to a single octal digit. Similarly, binary numerals can be converted to hexadecimal by grouping the binary digits in groups of four beginning at the ones place, with each group of four binary digits converting to a single hexadecimal digit.
Decimal  Binary  Octal  Hexadecimal 

0  0  0  0 
1  1  1  1 
2  10  2  2 
3  11  3  3 
4  100  4  4 
5  101  5  5 
6  110  6  6 
7  111  7  7 
8  1000  10  8 
9  1001  11  9 
10  1010  12  A 
11  1011  13  B 
12  1100  14  C 
13  1101  15  D 
14  1110  16  E 
15  1111  17  F 
16  10000  20  10 
17  10001  21  11 
20  10100  24  14 
25  11001  31  19 
32  100000  40  20 
40  101000  50  28 
49  110001  61  31 
63  111111  77  3F 
99  1100011  143  63 